19h

comment 
Ref request: If the natural density (relative to the primes) exists, then the Dirichlet density also exists, and the two are equal
(...) we can conclude that the statement was at least known (to Artin and contemporaries) as of July 1927. Unfortunately, I don't have a copy of Hasse's lectures on class field, but will look for them tomorrow. As for Landau, I can't read German, yet it's possible that the English 1986 edition of his Elementary Number Theory or the 1969 anthology of his papers by Bombieri and Davenport will do the job. Thanks for the pointers! 
19h

comment 
Ref request: If the natural density (relative to the primes) exists, then the Dirichlet density also exists, and the two are equal
@FranzLemmermeyer. I was hoping for a comment of yours. In the English translation of the 19231958 correspondence between Artin and Hasse, you note (p. 118): "When Artin writes that Chebotarev’s density theorem is "less precise" then he probably refers to the fact that Artin had used the natural density, whereas Chebotarev stated his result using the Dirichlet density. Any set of primes with a natural density also has a Dirichlet density (and both of them are equal), but the converse need not hold." So, if your interpretation of Artin's comment is correct, and it seems very plausible, (...) 
19h

comment 
Ref request: If the natural density (relative to the primes) exists, then the Dirichlet density also exists, and the two are equal
Incidentally, could you please provide a page for Hasse's remark? As I can't read German, this would help me much. 
19h

comment 
Ref request: If the natural density (relative to the primes) exists, then the Dirichlet density also exists, and the two are equal
Very interesting, thank you! I won't have access to Bateman's review until tomorrow, but there is a counterexample to Hasse's claim that is by now relatively wellknown: Serre (on p. 76 of the 1996 edition of A Course in Arithmetic) credits it to a private communication by Bombieri, and more details can be found in: D. I. A. Cohen and T. M. Katz, Prime Numbers and the First Digit Phenomenon, J. Number Theory 18 (1984), 261268. 
20h

accepted  Ref request: If the natural density (relative to the primes) exists, then the Dirichlet density also exists, and the two are equal 
22h

comment 
Ref request: If the natural density (relative to the primes) exists, then the Dirichlet density also exists, and the two are equal
Now I feel sorry for not having mentioned it in the OP, but I did already know all of this (my question is motivated by ideas matured in the shadow of Tenenbaum's section on densities). And I agree with you about the substantial difference in the difficulty levels of Theorems 2 and 3. However, I'm not convinced that a generalization of Theorem 3 to relative densities (if true!) would be that straightforward, while this is certainly the case with Theorem 2. 
1d

revised 
Ref request: If the natural density (relative to the primes) exists, then the Dirichlet density also exists, and the two are equal
Added a comment to clarify, I hope, the reasons why I'm asking 
1d

revised 
Ref request: If the natural density (relative to the primes) exists, then the Dirichlet density also exists, and the two are equal
Added a comment to clarify, I hope, the reasons why I'm asking 
1d

comment 
Ref request: If the natural density (relative to the primes) exists, then the Dirichlet density also exists, and the two are equal
@Lucia. Agree. Yet, I can't find any explicit occurrence of the same remark in any other place (before Serre's book), which sounds strange to me, all the more that I happen to know people attributing it to him. Also, the proof you refer to (by partial summation only), while being so simple, results into an inequality that is sensibly coarser than what can be actually proved. That's why I'd like to see how the proof has been written down, if it ever was. 
1d

revised 
Ref request: If the natural density (relative to the primes) exists, then the Dirichlet density also exists, and the two are equal
added 38 characters in body 
1d

asked  Ref request: If the natural density (relative to the primes) exists, then the Dirichlet density also exists, and the two are equal 
Jan
30 
comment 
Density of ratios of an arbitrary increasing sequence of prime numbers
Incidentally, an older result by T. Šalát (On ratio sets of sets of natural numbers, Acta Arith. 15 (1969), No. 3, 273278; Corrigendum to the paper ``On ratio sets of sets of natural numbers'', ibid. 16 (1969), 103) suggests as a starting point for further investigations to prove/disprove that if $μ^\ast(X)=\mu_\ast(X)>0$ or $\mu^\ast(X) = 1$ for $X \subseteq \bf P$ (I'm continuing with the notation used in my previous comments), then the ratio set of $X$ is dense in ${\bf R}^+$. 
Jan
30 
comment 
Density of ratios of an arbitrary increasing sequence of prime numbers
@GregMartin. Isn't $\mu^\ast(X)=\mu_\ast(X)=0$ if $X < \infty$? Perhaps it's me who is missing something. 
Jan
30 
comment 
Density of ratios of an arbitrary increasing sequence of prime numbers
$\mu_\ast$ is the conjugate of $\mu^\ast$, i.e. the fnc $\mathcal P({\bf P})\to{\bf R}:X\mapsto 1−\mu^\ast({\bf P}\setminus X)$. One reason why this is tempting is that the analogous result with the upper asymptotic density in place of $\mu^\ast$ (and its conjugate in place of $\mu_\ast$) is, in fact, true, see O. Strauch and J. T. Tóth, Asymptotic density of $A \subseteq \mathbf N$ and density of the ratio set $R(A)$, Acta Arith. 87 (1998), No. 1, 6778. 
Jan
30 
comment 
Density of ratios of an arbitrary increasing sequence of prime numbers
As you are are seeking sensible conditions etc., it seems reasonable to me that an answer to your question should be phrased in the "language of densities", but I don't think there is much known out there in this sense. E.g., let $\mu^\ast$ be the asymptotic density relative to $\bf P$ (= the set of primes), viz. the fnc $\mathcal P({\bf P})\to{\bf R}:X\mapsto\limsup_n\frac{1}{\pi(n)}X\cap[1,n]$, where $\pi$ is the prime counting fnc. Then, it's tempting to conjecture that for $X\subseteq\bf P$ the ratio set of $X$ is dense in ${\bf R}^+$ if $\mu^\ast(X)+\mu_\ast(X) \ge 1$, where (...) 
Jan
30 
comment 
Density of ratios of an arbitrary increasing sequence of prime numbers
I think Chris Ramsey means it's obvious the answer to your question is in the negative. E.g., start with $p_1 := 2$ and recursively define $p_{n+1}$ in such a way that $p_{n+1} \ge 2p_n$ (which is always possible, of course). Then $A(\mathbf p) \subseteq \{1\} \cup {]0,1/2]} \cup [2,\infty[$. (I'm afraid the question is not appropriate for MO.) 
Jan
30 
comment 
Reference request: Models of isomorphic languages result into isomorphic categories
@François. I hadn't realized it was a paper (sic!). But then, why don't you post your comment as an answer? It's much more than what I was looking for (which means a plus to me, not a minus). #Andrea. Yes, the claim is precisely the one at the end of your last comment. But nothing similar is discussed in Johnstone's Elephant, AFAICS. 
Jan
30 
comment 
Reference request: Models of isomorphic languages result into isomorphic categories
Unfortunately, I don't have Hájek's book and can't read German, but isn't the book about categorical logic (or, as some authors call it, cat model theory)? If so, then a better ref would be Sect. D in Johnstone's Elephant (see Peter's comment above). Now, it's a fact that model theory is a "fragment" of categorical logic: In the words of Makkai and Paré (from the introduction of their book), model theory is ``the study of ordinary, "setvalued", or ensembliste models of logical theories.'' So yes, my question could also be framed in the language of cats (and in a much more general form). 
Jan
30 
comment 
Reference request: Models of isomorphic languages result into isomorphic categories
@François. The link provides a notion of 'interpretation of a structure within another structure', while I'm using a locution of the form 'a model of a language $L=(\sigma, \Xi)$ of type $\mathscr{L}_\infty$' to mean a $\sigma$structure that satisfies the axioms in $\Xi$. Here, $\sigma$ is the signature of $L$ and $\Xi$ a (possibly empty) set of wffs derived from $\sigma$ and the (logical and nonlogical) symbols of $\mathscr{L}_\infty$ according to the formation rules of the extended firstorder logic specified in the OP, and 'to satisfy' just means what you guess. 
Jan
30 
comment 
Reference request: Models of isomorphic languages result into isomorphic categories
(...) (in the sense of categorical model theory) of an $\mathscr{L}_{\infty}$ language and models of a sketch give rise to equivalent categories (something stronger is true, but never mind). This fact is not related to the OP, yet it answers another question that I would have liked to ask for a while, so I thought to record it here. 